1 00:00:00,000 --> 00:00:05,812 People have thought, people have wondered about infinity for thousands of years. 2 00:00:05,812 --> 00:00:11,846 We're going to continue that tradition of thinking about infinity, but we're going 3 00:00:11,846 --> 00:00:16,555 to focus in on one specific instance of infinity in mathematics. 4 00:00:16,555 --> 00:00:21,706 What does it mean to say that the limit of a function equals infinity? 5 00:00:21,706 --> 00:00:26,047 Here's what it means. The limit of F of X, as X approaches A 6 00:00:26,047 --> 00:00:30,315 equals infinity means that F of X is as large as you like. 7 00:00:30,315 --> 00:00:35,186 Provided x is close enough to A. As a bit of a warning, I'm not saying 8 00:00:35,186 --> 00:00:39,880 that this number limit F of X, X approaches A is equal to this number, 9 00:00:39,880 --> 00:00:44,030 infinity is not a number. What I am doing here is attaching a 10 00:00:44,030 --> 00:00:49,267 precise meaning to the entire statement that the limit of F of X is infinity. 11 00:00:49,267 --> 00:00:53,893 So, that's what it means precisely, but what does it mean practically. 12 00:00:53,893 --> 00:00:56,677 Let's go to the board. This is an example, 13 00:00:56,677 --> 00:01:00,268 the limit of one over X squared as X approaches zero. 14 00:01:00,268 --> 00:01:05,688 Now this limit is not equal to a number. One over X squared is not getting close 15 00:01:05,688 --> 00:01:10,837 to a number when X is approaches zero. But one over X squared is as big as I 16 00:01:10,837 --> 00:01:14,360 want it to be if I'm willing to put X close to zero. 17 00:01:14,360 --> 00:01:19,884 For instance, if X is within a 1000th of zero, what happens? 18 00:01:19,884 --> 00:01:27,313 Well to say X is within a 1000th of zero is to say that X is between minus.00 19 00:01:27,313 --> 00:01:31,694 and.001. Now, X is also non zero here because I 20 00:01:31,694 --> 00:01:38,171 don't want to divide by zero. So I've got a non zero number within a 21 00:01:38,171 --> 00:01:42,118 1000th of zero. What does that mean about X squared? 22 00:01:42,118 --> 00:01:45,406 [SOUND]. Well that means that X squared is, it's 23 00:01:45,406 --> 00:01:50,722 bigger than zero because if I square any non zero number, it's positive and it's 24 00:01:50,722 --> 00:01:55,759 less than, [SOUND] a millionth. So if I take a number within a thousandth 25 00:01:55,759 --> 00:01:59,957 of zero and I square it, that was within a millionth of zero. 26 00:01:59,957 --> 00:02:03,944 And [SOUND] what does that mean about one over X squared? 27 00:02:03,944 --> 00:02:09,121 Well if X squared is within a millionth of zero, one over X squared is now 28 00:02:09,121 --> 00:02:12,990 bigger, [SOUND] than a million. What happened, right? 29 00:02:12,990 --> 00:02:17,770 I made one over x squared very large by making x close enough but not equal to 30 00:02:17,770 --> 00:02:20,553 zero. And there's nothing particularly special 31 00:02:20,553 --> 00:02:24,002 about these numbers. If I wanted one over x squared to be 32 00:02:24,002 --> 00:02:28,601 bigger than a trillion, I would just need x to be within a millionth of zero. 33 00:02:28,601 --> 00:02:32,474 This example is at once maybe too complicated because I've got some 34 00:02:32,474 --> 00:02:36,528 explicit numbers in here. And maybe this formula is not complicated 35 00:02:36,528 --> 00:02:39,614 enough to, to really get a sense of what's going on. 36 00:02:39,614 --> 00:02:42,700 Let's look at a slightly more complicated function. 37 00:02:45,280 --> 00:02:51,691 So what's the limit of X plus two over X squared minus 2x plus one as X approaches 38 00:02:51,691 --> 00:02:54,195 one. And this is a limit of a quotient. 39 00:02:54,195 --> 00:02:58,600 So your first temptation is to think the limit of a quotient the quotients the 40 00:02:58,600 --> 00:03:01,220 limits. But what's the limit of the denominator? 41 00:03:01,220 --> 00:03:04,510 The denominator is a polynomial. Polynomials are continuous. 42 00:03:04,510 --> 00:03:08,468 So I can evaluate the limit of the denominator just by plugging in one. 43 00:03:08,468 --> 00:03:11,256 And if I plug in one, I get one minus two plus one. 44 00:03:11,256 --> 00:03:13,988 That's zero. The limit of the denominator is zero. 45 00:03:13,988 --> 00:03:18,225 So I can't simply say that the limit of the quotient is the quotient of the 46 00:03:18,225 --> 00:03:21,570 limits in this case. Because the limit of the denominator is 47 00:03:21,570 --> 00:03:24,191 zero. And that limit law does not apply in this 48 00:03:24,191 --> 00:03:26,034 case. What am I going to do? 49 00:03:26,034 --> 00:03:30,070 Well I could also notice the denominator factors. 50 00:03:30,070 --> 00:03:33,942 X squared minus 2X plus one, I can rewrite that. 51 00:03:33,942 --> 00:03:38,801 [SOUND] X squared minus 2X plus one is X minus one squared. 52 00:03:38,801 --> 00:03:45,226 [SOUND] So instead of evaluating this limit, I could try to look at this limit, 53 00:03:45,226 --> 00:03:50,910 the limit of X plus two over X minus one squared as X approaches one. 54 00:03:50,910 --> 00:03:56,280 Now what do I know? The numerator is X plus two and if X is 55 00:03:56,280 --> 00:04:00,930 close to one, I can make X plus two close to three. 56 00:04:00,930 --> 00:04:07,384 What about the denominator? A number close to one, minus one squared, 57 00:04:07,384 --> 00:04:13,357 is a number close to zero. But not just a number close to zero. 58 00:04:13,357 --> 00:04:19,427 It's a positive number close to zero. So let me write this down. 59 00:04:19,427 --> 00:04:23,762 Alright. The numerator [SOUND] is about three. 60 00:04:23,762 --> 00:04:28,579 [SOUND] And the denominator [SOUND] is about what? 61 00:04:28,579 --> 00:04:32,529 Well, it's some small but positive number. 62 00:04:32,529 --> 00:04:35,745 [SOUND]. And what happens if I take a number, 63 00:04:35,745 --> 00:04:40,499 like, near three, and divide it by a number which is small and positive? 64 00:04:40,499 --> 00:04:45,797 That number can be as big as I like. So I can make this quantity as large as I 65 00:04:45,797 --> 00:04:50,646 like, if I make x close enough to one. Because I can make the numerator close to 66 00:04:50,646 --> 00:04:54,929 three, and I can make the denominator as close to zero and positive as I like. 67 00:04:54,929 --> 00:04:59,379 And if I take a number close to three and divide it by a number close to zero, I 68 00:04:59,379 --> 00:05:03,329 can make a number as large as I want. So this limit [SOUND] is equal to 69 00:05:03,329 --> 00:05:07,526 infinity. The homework includes even trickier 70 00:05:07,526 --> 00:05:12,352 situations. For instance, limit problems where x 71 00:05:12,352 --> 00:05:17,383 approaches negative infinity, instead of infinity. 72 00:05:17,383 --> 00:05:22,209 If you get stuck I encourage you to contact us. 73 00:05:22,209 --> 00:05:24,571 We're here to help you.